A main source of gene expression noise in prokaryotes is translational bursting. It arises from efficient translation of mRNAs with low copy numbers, which makes the production of protein copies highly variable and pulsatile. To obtain analytical solutions, previous models to capture this noise source had to assume translation to be initiation-limited, representing the burst size by a specific type of a long-tail distribution. However, there is increasing evidence suggesting that the initiation is not the rate-limiting step in certain settings, for example, under stress conditions. Here, to overcome the limitations imposed by the initiation-limited assumption, we present a new analytical approach that can evaluate biological consequences of the protein burst size with a general distribution. Since our new model can capture the contribution of other factors to the translational noise, it can be used to analyze the effects of gene expression noise in more general settings. We used this new model to analytically analyze the connection between the burst size and the stability of gene expression processes in various settings. We found that the burst size with different distributions can lead to quantitatively and qualitatively different stability characteristics of protein abundance and can have non-intuitive effects. By allowing analysis of how the stability of gene expression processes changes based on various distributions of translational noise, our analytical approach is expected to enable deeper insights into the control of cell fate decision-making, the evolution of cryptic genetic variations, and fine-tuning of gene circuits.